Evolution of 2d Potts Model Grain Microstructures from an Initial Hillert Size Distribution

Grain growth experiments and simulations exhibit self-similar grain size distributions quite different from that derived via a mean field approach by Hillert [ 11. To test whether this discrepancy is due to insufficient anneal times, two different two-dimensional grain structures with realistic topologies and Hillert grain size distributions are generated and subjected to grain growth via the Monte Carlo Potts Model (MCPM). In both cases, the observed self-similar grain size distributions deviate from the initial Hillert form and conform instead to that observed in MCPM grain growth simulations that start from a random microstructure. This suggests that the Hillert grain size distribution is not an attractor. Introduction It has long been observed that the grain growth process in polycrystalline materials is self-similar. The grain size distribution, scaled by the average grain size, is constant over time so that a coarsened structure is statistically equivalent to a magnification of its antecedent structure. Smith wrote in 1952 that “there is probably a tendency toward a fixed distribution of shapes and relative cell sizes ...” [2] The nature of this distribution has been a topic of debate since then. Several mean field models for grain growth begin from an equation for the flux of grains as a function of the grain size and time, 8 j = D+ f(R,t) v, aR where D is a diffusion coefficient for grains in grain size-time space, v is the boundary velocity term on the right hand side of Equ. 1 is the diffusion term; the second is the drift term. and setting v to a physically motivated but non-rigorous growth function containing a critical grain radius R‘; Lucke et al. later justified Hillert’s v more rigorously [3]. According to Hillert’s analysis, R/ t , and f(R, t) is the grain size distribution as a function of grain radius R and time t. The first Hillert [ 11 solved for the presumably self-similarf(R, t) by ignoring the diffusion term, ’ where d is the dimensionality, k = R/Rc, and RL is a fitting parameter [ 13. Several corrections and modifications to Hillert’s analysis have been proposed, but none entail dramatic changes to f(R,t) for normal grain growth [3,4]. In general, the Hillert distribution is more sharply peaked and skewed to larger R than experimental data [5]. In addition, various computer simulations of normal grain growth have produced distributions which appear self-similar over time and which have the characteristics of the experimental distributions [5-SI. These discrepancies have been attributed by some to the notion that the experiments and simulations span anneal times that are too short to allow a transition to the Hillert distribution. beyond their current maxima. Therefore, it is difficult to determine whether such simulations that obey the Hillert grain size distribution, and observe their evolution under normal grain growth conditions. Presumably, if the Hillert distribution is an attractor, these structures should evolve self-similarly from the start, maintaining the Hillert grain size distribution as they grow. It is not computationally tractable to increase grain growth simulation times significantly :